Inference on conditional quantile processes in partially linear models with applications to the impact of unemployment benefits
Qu, Zhongjun; Yoon, Jungmo; Perron, Pierre
We propose methods to estimate and conduct inference on conditional quantile processes for models with nonparametric and linear components. The estimation procedure uses local linear or quadratic regressions, with the bandwidth allowed to vary across quantiles to adapt to data sparsity. We establish a Bahadur representation that holds uniformly in the covariate value and the quantile index. Then,we show that the proposed estimator converges weakly to a Gaussian process and develop methods for constructing uniform confidence bands and hypothesis testing. Our results also cover locally partially linear models with boundary points, thereby allowing for Sharp Regression Discontinuity Designs (SRD). This allows us to study the effects of unemployment insurance (UI) benefits extensions using the dataset of Nekoei and Weber (2017) who found a statistically significant effect, though of minor economic importance using an SRD focusing on the average effect. Our model allows heterogeneity with respect to both the covariate and the quantile. We find economically strong significant effects in the tail of the distribution,say the 10% quantile of the outcome variable (e.g., the wage change distribution). Under a rank invariance assumption, this implies that individuals who benefited the most are those who would have experienced substantial wage cuts if there were no benefit extension. Since our setup allows for discrete covariates, we also find positive and statistically significant effects for white-collar and female workers and those with a college education, but not for blue-collar male workers without higher education. Hence, while UI benefits reduce the within-group inequality for some subgroups by covariates, they can be viewed as regressive and enhancing between-group inequality, although they also help to bridge the gender gap.
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